Regular and Non-regular Polygon Areas

Date: 03/10/99 at 17:10:37
From: Robert Davies
Subject: Regular Polygons
I am trying to work out the proof that regular polygons give the
maximum area but as of yet have not succeeded.
Please help!

Date: 03/11/99 at 11:55:35
From: Doctor Peterson
Subject: Re: Regular Polygons
The first approach that comes to mind is to take any non-regular
polygon and show that you can find a larger polygon with the same
perimeter. Then the largest polygon with that perimeter must be
regular.
Suppose that there are three consecutive vertices A, B, and C in a
polygon such that AB and BC have different lengths. See if you can find
a point B' where AB' and B'C are the same length, but their sum is the
same as AB + BC. Then show that the area of triangle AB'C will be
larger than that of ABC. If you replace B with B' in the polygon, its
perimeter stays the same but the area is larger.
B'
B +
+ / \
/ \ / \
/ / \ \
/ / \ \
/ / \ \
/ / \ \
/ / \ \
A // \\ C
+ - - - - - - - - - - - - - - - - - +
| \
| \
... ...
This will show that the largest polygon has to have all sides the same,
since a polygon whose sides are not the same is never the largest.
You will also have to show that the angles in the largest polygon with
a given perimeter are all the same. Try a method similar to what we
just did for the sides. (I have not taken the time to work that part
out.)
- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/